Pour the warm milk into a large mixing bowl or the bowl of a stand mixer. Sprinkle the active dry yeast and 1 tablespoon of the granulated sugar over the top. Let it sit for 5–10 minutes, until the mixture looks foamy and creamy. If it doesn’t foam, discard and start again with fresh yeast.

Once the yeast mixture is foamy, whisk in the remaining granulated sugar, eggs, melted butter, and vanilla extract. The mixture may look slightly lumpy from the butter, which is fine.

In a separate bowl, whisk together 3 1/2 cups of the flour and the salt. Add this flour mixture to the wet ingredients about 1 cup at a time, stirring with a wooden spoon or using the dough hook on low speed. The dough should start to pull away from the sides but remain soft and slightly tacky. Add up to 1/2 cup more flour only if the dough is too wet to handle.

Turn the dough out onto a lightly floured surface and knead for 5–7 minutes, until smooth, elastic, and slowly springing back when poked. If using a stand mixer, knead with the dough hook on medium-low speed for 4–5 minutes. Avoid adding too much extra flour; the dough should remain soft, not stiff.

Shape the dough into a ball and place it in a lightly oiled bowl, turning once to coat. Cover with a clean kitchen towel or plastic wrap and let rise in a warm, draft-free spot for 1–1 1/2 hours, or until doubled in size.

Gently deflate the risen dough, cover the bowl again, and refrigerate for at least 1 hour or up to overnight. This develops flavor and makes the dough easier to roll. If short on time, you can skip this chill and proceed.

Lightly flour your work surface. Turn out the dough and roll it into a rectangle about 1/4 inch thick, keeping the thickness as even as possible. If the dough springs back, let it rest for 5 minutes and roll again.

Using a pizza cutter or sharp knife, cut the dough into roughly 2 1/2-inch squares or rectangles. Transfer the pieces to a lightly floured surface or tray and cover with a clean towel while you heat the oil so they don’t dry out.

Pour neutral oil into a heavy-bottomed pot or Dutch oven to a depth of 2–3 inches. Attach a deep-fry thermometer if you have one. Heat the oil to 360–370°F, adjusting the heat as needed to maintain temperature during frying.

Working in small batches of 4–6 pieces, carefully slip the dough squares into the hot oil, away from you. They should puff within 30–40 seconds. Fry for about 1–1 1/2 minutes per side, turning once, until deep golden. Use a slotted spoon or spider to transfer to a paper towel–lined tray to drain.

While the beignets are still hot, generously sift powdered sugar over them, or place a few at a time into a paper bag with powdered sugar and gently shake to coat.

Serve the beignets immediately, ideally within 15–20 minutes of frying, with extra powdered sugar on the side and optional cinnamon, chocolate sauce, or café au lait for dipping.